Nondegenerate wavefunctions

Postulate VII. If, for a given system, the wavefunction 4 is a linear combination of nondegenerate wavefunctions 4 that have eigen values a ... 

For nondegenerate vibrations all symmetry operations change Qj into 1 times itself. Hence Q/ is unchanged by all symmetry operations. In other words, Q and consequently y(O) behave as totally symmetric functions (i.e. the function is independent of symmetry). However, the wavefunction of the first excited state 3(1) has the same symmetry as Qj. For example, the wavefunction of a totally symmetric vibration (e.g. Qi of C02) is itself a totally symmetric function. 

If the initial ground-state wavefunction (/(q is nondegenerate, the first-order term (i. e., the second term) in Eq. (1) is nonzero only for the totally-symmetrical nuclear displacements (note that g, and (dH/dQi) have the same symmetry). Information about the equilibrium nuclear configuration after the symmetrical first-order deformation will be given by equating the first-order term to zero. 

A rigorous modelling of thermal broadening is — in practice — quite cumbersome and tedious. Let us consider a general asymmetric top molecule such as H2O, for example. Each total angular momentum state, specified by the quantum number J, splits into (2 J + 1) nondegenerate substates with energies E 0f (K = 1,..., 2J + 1). Every one of these (2J + 1) rotational states corresponds to a different type of rotational motion and is described by a distinct rotational wavefunction (see Section 11.3). 

Each nondegenerate eigenstate has a distinct wavefunction which is uniquely characterized by the totality of expansion coefficients G jp) in Equation (11.10). 

Any nonzero volume piece pd(r) of the nondegenerate ground state electron density fully determines the ground state energy E, the ground state wavefunction P (up to a phase factor), and the expectation values of all spin-free operators defined by the ground state wavefunction VP . 

Lemma 1. Defining the set V of external potentials v xti ) leading to a nondegenerate ground state and S the set of wavefunctions S (r. .., r r) which are ground state wavefunctions of a system of N electrons in an external potential, the application ... 

If the unit cell of the crystal contains only one molecule, and the molecular term is nondegenerate, the wavefunction (2.8) takes the form... 

Let us now pick an arbitrary density out of the set A of densities of nondegenerate ground states. The Hohenberg-Kohn theorem then tells us that there is a unique external potential v (to within a constant) and a unique ground state wavefunction I W[ri]) (to within a phase factor) corresponding to this density. This also means that the ground state expectation value of any observable, represented by an operator O. can be regarded as a density functional... 

Theorem 1 (Hohenberg-Kohn). The density n corresponding to a nondegenerate ground state specifies the external potential v up to a constant and the ground state wavefunction I W[n]) up to a phase factor. Moreover,... 

Note that these coefficients are well-defined because Ek > E0, since we are dealing with an isolated nondegenerate ground state. We therefore find for the first order change in the wavefunction... 

Nesbet considers a system of N noninteracting electrons with a nondegenerate ground state and with the wavefunctions in the form of Slater determinants = l/ /A Det . [Pg.99]

In the treatment here we have assumed that Ey is a ground-state wavefunction of a Hamiltonian Hv corresponding to the ground-state density pv, but no further assumption has been made. The results hold for nondegenerate as well as degenerate ground states. 

An important extension of the original Hohenberg-Kohn approach has been proposed by Levy [14, 15] based on earlier work by Percus [16]. The functional F[p] of Hohenberg and Kohn is defined only for densities which are obtained from a nondegenerate ground-state wavefunction corresponding to an external local potential. Levy introduced a functional... 

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